Affiliations: 1: Department of Population Health and Pathobiology, College of Veterinary Medicine, North Carolina State University, Raleigh, NC 27607;
2: Department of Population Health and Pathobiology, College of Veterinary Medicine, North Carolina State University, Raleigh, NC 27607;
3: Department of Animal and Food Science, University of Delaware, Newark, DE;
4: North Carolina State University, College of Veterinary Medicine, Raleigh, NC

This article provides an overview of the emerging field of mathematical modeling in preharvest food safety. We describe the steps involved in developing mathematical models, different types of models, and their multiple applications. The introduction to modeling is followed by several sections that introduce the most common modeling approaches used in preharvest systems. We finish the chapter by outlining potential future directions for the field.

20.Jordan D, Nielsen LR, Warnick LD. 2008. Modelling a national programme for the control of foodborne pathogens in livestock: the case of Salmonella Dublin in the Danish cattle industry. Epidemiol Inf136:1521–1536. [PubMed][CrossRef]

This article provides an overview of the emerging field of mathematical modeling in preharvest food safety. We describe the steps involved in developing mathematical models, different types of models, and their multiple applications. The introduction to modeling is followed by several sections that introduce the most common modeling approaches used in preharvest systems. We finish the chapter by outlining potential future directions for the field.

An example of a deterministic compartmental infectious disease model. (A) The flowchart represents an SIRS model. Compartments S, I, and R track the number of individuals who are susceptible, infectious, and recovered, respectively. The triangles represent change in the number of individuals in the compartments. Models that contain both demographic (e.g., births and deaths) and infection flows are called endemic models. (B) Compartmental models are often described mathematically as ordinary differential equations. The ordinary differential equations here describe the inflows and outflows by which the number of individuals in each epidemiological state (S, I, and R) changes over time. The total population, N is equal to S + I + R; υ = birth and death rate; γ = recovery rate of infectious individuals; β = transmission coefficient (transmission is frequency dependent); and r = immunity loss rate. The inflow and outflow of a compartment is reflected in the equation. For instance, the S compartment has four arrows associated with it: two inflows and two outflows. The two inflows are birth and loss of immunity from recovered compartment (hence the + sign in the first equation); the two outflows are natural death and infection to infectious compartment (hence the – sign in the first equation). (C) The basic reproduction number of this model.

microbiolspec/4/4/PFS-0001-2013-fig4_thmb.gif

microbiolspec/4/4/PFS-0001-2013-fig4.gif

FIGURE 4

An example of a deterministic compartmental infectious disease model. (A) The flowchart represents an SIRS model. Compartments S, I, and R track the number of individuals who are susceptible, infectious, and recovered, respectively. The triangles represent change in the number of individuals in the compartments. Models that contain both demographic (e.g., births and deaths) and infection flows are called endemic models. (B) Compartmental models are often described mathematically as ordinary differential equations. The ordinary differential equations here describe the inflows and outflows by which the number of individuals in each epidemiological state (S, I, and R) changes over time. The total population, N is equal to S + I + R; υ = birth and death rate; γ = recovery rate of infectious individuals; β = transmission coefficient (transmission is frequency dependent); and r = immunity loss rate. The inflow and outflow of a compartment is reflected in the equation. For instance, the S compartment has four arrows associated with it: two inflows and two outflows. The two inflows are birth and loss of immunity from recovered compartment (hence the + sign in the first equation); the two outflows are natural death and infection to infectious compartment (hence the – sign in the first equation). (C) The basic reproduction number of this model.

Network models are usually represented as graphs. A graph contains a set of nodes (individuals) and edges (contacts) that are associated with these nodes. Shown is an example with five interacting calves. The calves with contact with each other have a line (edge) between them to indicate their mutual relationship (A). The graph can be translated to an adjacency matrix, which contains as many rows and columns as there are nodes (B). The elements of the matrix record information about the edges between each pair of nodes, with 1 indicating a contact and 0 meaning no contact.

microbiolspec/4/4/PFS-0001-2013-fig5_thmb.gif

microbiolspec/4/4/PFS-0001-2013-fig5.gif

FIGURE 5

Network models are usually represented as graphs. A graph contains a set of nodes (individuals) and edges (contacts) that are associated with these nodes. Shown is an example with five interacting calves. The calves with contact with each other have a line (edge) between them to indicate their mutual relationship (A). The graph can be translated to an adjacency matrix, which contains as many rows and columns as there are nodes (B). The elements of the matrix record information about the edges between each pair of nodes, with 1 indicating a contact and 0 meaning no contact.